()) If $$ \cos ( heta)=-\frac{8}{17} $$ and $$ 90^{\circ}

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()) If $$ \cos (	heta)=-\frac{8}{17} $$ and $$ 90^{\circ}<	heta<180^{\circ} $$, what is $$ \sin (	heta) $$ ? $$ \sin (	heta)= $$ Write your answer in simplified, rationalized form.

UpStudy AI · Accepted Answer

Given that $$ \cos(\theta) = -\frac{8}{17} $$ and $$ 90^{\circ} < \theta < 180^{\circ} $$, we can determine $$ \sin(\theta) $$ using the Pythagorean identity:

$$
\sin^2(\theta) + \cos^2(\theta) = 1
$$

Substituting the given value of $$ \cos(\theta) $$:

$$
\sin^2(\theta) + \left(-\frac{8}{17}\right)^2 = 1
$$

$$
\sin^2(\theta) + \frac{64}{289} = 1
$$

$$
\sin^2(\theta) = 1 - \frac{64}{289} = \frac{225}{289}
$$

Taking the square root of both sides:

$$
\sin(\theta) = \sqrt{\frac{225}{289}} = \frac{15}{17}
$$

Since $$ \theta $$ is in the second quadrant, where sine is positive, we have:

$$
\sin(\theta) = \frac{15}{17}
$$

**Answer:** $$ \boxed{\dfrac{15}{17}} $$
$$ \sin(\theta) = \frac{15}{17} $$

()) If and , what is ?
Write your answer in simplified, rationalized form.

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